Time-dependent Nonlinear Partial Differential Equations Research Articles

Hybrid OpenMP/MPI programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

We present hybrid OpenMP/MPI (Open Multi-Processing/Message Passing Interface) parallelized versions of earlier published C programs (D. Vudragovic et al., Comput. Phys. Commun. 183, 2021 (2012)) for calculating both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation in three spatial dimensions. The GP equation describes the properties of dilute Bose-Einstein condensates at ultra-cold temperatures. Hybrid versions of programs use the same algorithms as the C ones, involving real- and imaginary-time propagation based on a split-step Crank-Nicolson method, but consider only a fully-anisotropic three-dimensional GP equation, where algorithmic complexity for large grid sizes necessitates parallelization in order to reduce execution time and/or memory requirements per node. Since distributed memory approach is required to address the latter, we combine MPI programing paradigm with existing OpenMP codes, thus creating fully flexible parallelism within a combined distributed/shared memory model, suitable for different modern computer architectures. The two presented C/OpenMP/MPI programs for real- and imaginary-time propagation are optimized and accompanied by a customizable makefile. We present typical scalability results for the provided OpenMP/MPI codes and demonstrate almost linear speedup until inter-process communication time starts to dominate over calculation time per iteration. Such a scalability study is necessary for large grid sizes in order to determine optimal number of MPI nodes and OpenMP threads per node.

CUDA programs for solving the time-dependent dipolar Gross–Pitaevskii equation in an anisotropic trap

In this paper we present new versions of previously published numerical programs for solving the dipolar Gross-Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time, yielding both stationary and non-stationary solutions. New versions of programs were developed using CUDA toolkit and can make use of Nvidia GPU devices. The algorithm used is the same split-step semi-implicit Crank-Nicolson method as in the previous version (R. Kishor Kumar et al., Comput. Phys. Commun. 195, 117 (2015)), which is here implemented as a series of CUDA kernels that compute the solution on the GPU. In addition, the Fast Fourier Transform (FFT) library used in the previous version is replaced by cuFFT library, which works on CUDA-enabled GPUs. We present speedup test results obtained using new versions of programs and demonstrate an average speedup of 12 to 25, depending on the program and input size.

Towards efficient backward-in-time adjoint computations using data compression techniques

Abstract In the context of a posteriori error estimation for nonlinear time-dependent partial differential equations, the state-of-the-practice is to use adjoint approaches which require the solution of a backward-in-time problem defined by a linearization of the forward problem. One of the major obstacles in the practical application of these approaches is the need to store, or recompute, the forward solution to define the adjoint problem and to evaluate the error representation. This study considers the use of data compression techniques to approximate forward solutions employed in the backward-in-time integration. The development derives an error representation that accounts for the difference between the standard-approach and the compressed approximation of the forward solution. This representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data-compressed reconstructed solution (i.e. scalar quantities that can be evaluated as the forward problem is integrated). This approach is then compared with existing techniques, such as checkpointing and time-averaged adjoints. Finally, we provide numerical results indicating the potential efficiency of our approach on a transient diffusion–reaction equation and on the Navier–Stokes equations. These results demonstrate memory compression ratios up to 450 × while maintaining reasonable accuracy in the error-estimates.

Space–time DG methods for the coupled electro–mechanical activation of the human heart

AbstractWe consider the coupled system of time–dependent nonlinear partial differential equations modeling the electromechanical response of human heart tissue. Instead of time–stepping schemes we use a discontinuous Galerkin finite element method in the space–time domain to be able to resolve the solution in space and time simultaneously. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Runge–Kutta Residual Distribution Schemes

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.

Two Efficient Approaches Based On Radial Basis Functions To Nonlinear Time-dependent Partial Differential Equations

In this paper, two numerical meshless approaches, based on radial basis functions (RBFs) are applied to solve nonlinear time-dependent partial differential equations. Both of these approaches are based on Kansa meshless method. In these procedures the time is descritized to small time steps, the space is also descritized in each sub-domain. Then by the Kansa collocation method, by using RBFs, the approximate solution, in each step, is obtained. In the first approach, the nonlinear terms are eliminated, by linearization. In the second approach nonlinear equations are solved directly, by Fix point method. Four examples are provided to illustrate the efficiency and the reliability of these approaches. The results are also compared with each other.

Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations

We propose an efficient Legendre-Gauss collocation algorithm forsecond-order nonlinear ordinary differential equations (ODEs). Wealso design a Legendre-Gauss-type collocation algorithm for time-dependent second-order nonlinear partialdifferential equations (PDEs), which can be implemented in asynchronous parallel fashion. Numerical results indicate the highaccuracy and effectiveness of the suggested algorithms.

Legendre–Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations

We propose new Legendre–Gauss collocation algorithms for ordinary differential equations. We also design Legendre–Gauss-type collocation algorithms for time-dependent nonlinear partial differential equations. The suggested algorithms enjoy spectral accuracy in both time and space, and can be implemented in a fast and stable manner. Numerical results exhibit the effectiveness.

Modeling of wave modes on a vertical film of a viscous ferromagnetic fluid flowing down a cylindrical electric conductor

The present study considers axially symmetrical waves on the surface of a ferromagnetic viscous fluid film flowing down a cylindrical conductor with a non-constant current. In addition to the gravitational force, the film is affected by a uniform magnetic field. Using the long-wave approximation, a model equation is obtained for the evolution of the film thickness from its undisturbed value. By showing that solutions with and without a constant magnetic field are self-similar, scaling parameters are identified and presented. Numerical solutions of the time-dependent non-linear partial differential equation are presented when current in the conductor is described by a ramp function (i.e., magnetic field onset).

A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations

A wavelet method is proposed for solving a class of nonlinear time-dependent partial differential equations. Following this method, the nonlinear equations are first transfoulled into a system of ordinary differential equations by using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. To justify the present method, the coupled viscous Burgers' equations are solved as examples, results demonstrate that the proposed wavelet algorithm have a much better accuracy and efficiency than many existing numerical methods, and the order of convergence of such a wavelet method can even reach about 5.

A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations

A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations

Fully implicit Lagrange-Newton-Krylov-Schwarz algorithms for boundary control of unsteady incompressible flows

SUMMARY We develop a parallel fully implicit domain decomposition algorithm for solving optimization problems constrained by time-dependent nonlinear partial differential equations. In particular, we study the boundary control of unsteady incompressible Navier–Stokes equations. After an implicit discretization in time, a fully coupled sparse nonlinear optimization problem needs to be solved at each time step. The class of full space Lagrange–Newton–Krylov–Schwarz algorithms is used to solve the sequence of optimization problems. Among optimization algorithms, the fully implicit full space approach is considered to be the easiest to formulate and the hardest to solve. We show that Lagrange–Newton–Krylov–Schwarz, with a one-level restricted additive Schwarz preconditioner, is an efficient class of methods for solving these hard problems. To demonstrate the scalability and robustness of the algorithm, we consider several problems with a wide range of Reynolds numbers and time step sizes, and we present numerical results for large-scale calculations involving several million unknowns obtained on machines with more than 1000 processors. Copyright © 2012 John Wiley & Sons, Ltd.

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

AbstractThis article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

The Use of Homotopy Analysis Method to Solve the Time-Dependent Nonlinear Eikonal Partial Differential Equation

In this research work a time-dependent partial differential equation which has several important applications in science and engineering is investigated and a method is proposed to find its solution. In the current paper, the homotopy analysis method (HAM) is developed to solve the eikonal equation. The homotopy analysis method is one of the most effective methods to obtain series solution. HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of a series solution. Furthermore, this method does not require any discretization, linearization or small perturbation and therefore reduces the numerical computation a lot. Some test problems are given to demonstrate the validity and applicability of the presented technique.

A moving mesh approach to an ice sheet model

A moving mesh approach to the numerical modelling of problems governed by nonlinear time-dependent partial differential equations (PDEs) is applied to the numerical modelling of glaciers driven by ice diffusion and accumulation/ablation. The primary focus of the paper is to demonstrate the numerics of the moving mesh approach applied to a standard parabolic PDE model in reproducing the main features of glacier flow, including tracking the moving boundary (snout). A secondary aim is to investigate waiting time conditions under which the snout moves.

The Use of Homotopy Analysis Method to Solve the Time-Dependent Nonlinear Eikonal Partial Differential Equation

The Use of Homotopy Analysis Method to Solve the Time-Dependent Nonlinear Eikonal Partial Differential Equation

RBFs Meshless Method of Lines for the Numerical Solution of Time-Dependent Nonlinear Coupled Partial Differential Equations

In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-difference and finite element methods to approximate spatial derivatives, this new technique does not require a mesh in the problem domain, and a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions. Accuracy of the method is assessed in terms of the error norms L2, L∞ and the three invariants C1, C2, C3. Numerical experiments are performed to demonstrate the accuracy and easy implementation of this method for the three classes of time-dependent nonlinear coupled partial differential equations.

Fully discrete linear approximation scheme for electric field diffusion in type-II superconductors

Superconductors are attracting physicists thanks to their ability to conduct electric current with virtually zero resistance. Their nonlinear behaviour opens, on the other hand, challenging problems for mathematicians. Our model of the diffusion of electric field in superconductors is based on three pillars: the eddy-current version of Maxwell’s equations, power law model of type-II superconductivity and linear dependence of magnetic induction on magnetic field. This leads to a time-dependent nonlinear degenerate partial differential equation. We propose a linear fully discrete approximation scheme to solve it. We have proven the convergence of the method and derived the error estimates describing the dependence of the error on the discretization parameters. These theoretical results were successfully confronted with numerical experiments.

A meshless method of lines for the numerical solution of KdV equation using radial basis functions

In this paper, a meshless method of lines (MOL) is presented for the numerical solution of the Korteweg–de Vries (KdV) equation. This novel method has an advantage over the traditional method of lines which approximates the spatial derivatives using finite difference method (FDM) or finite element method (FEM), because it does not need the mesh in the domain, and it approximates the solution using the radial basis functions (RBFs) on a set of node scattered in problem domain. A comparison among some RBFs is made in numerical examples. Numerical examples demonstrate the accuracy and easy implementation of this novel method and it is an efficient method for the nonlinear time-dependent partial differential equations (PDEs).

On adaptive time stepping for large-scale parabolic problems: Computer simulation of heat and mass transfer in vacuum freeze-drying

The work is motivated by the problem of freeze-drying, which is a process of dehydrating frozen materials by sublimation under high vacuum. In particular, it concerns the mathematical modelling and computer simulation of the heat and mass transfer with the core in solving the time-dependent nonlinear partial differential equation of parabolic type. Instead of a uniform discretization of the considered time interval, an adaptive time-stepping procedure is applied in an effort to optimize the whole simulation. The procedure is based on the local comparison of the Crank–Nicolson and backward Euler approximations. The results of numerical experiments performed on a selected real-life problem are included.